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Structural Decomposition
Methods of CSPs
• Input: a constraint hypergraph
• Output: an equivalent join tree
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A constraint hypergraph Hcg
associated with a CSP
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An equivalent join tree of Hcg
Criteria to compare structural decomposition methods:
(1) CPU time for decomposition (2) Width of the join tree
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
1
Contribution in Context
HYPERTREE[1]
TRAVERSE
CaT
CUT
HINGE+
HINGETCLUSTER[5]
HYPERCUTSET[11]
HINGE[5] TCLUSTER[8]  w*[6]
= TREEWIDTH [9]
BICOMP[3]
CUTSET[4]
HINGE+: An improvement to HINGE.
CUT: A variation of HINGE+.
TRAVERSE: Based on a simple sweep of the constraint hypergraph.
CaT: A combination of CUT and TRAVERSE.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
2
i-cut
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A 2-cut: {S6, S12}.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
3
Hinge decomposition (HINGE)
HINGE
Input:
A constraint hypergraph H.
Output: An equivalent join tree T of H.
The edges of T are labeled.
Process: Continuously finds 1-cuts in H.
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Applying
HINGE to
Hcg
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Width = 12
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
4
Hinge+ Decomposition (HINGE+)
HINGE+
Input:
a constraint hypergraph H and a maximum cut size k.
Output: An equivalent join tree T of H. The edges of T are labeled.
Process: Finds 1-cuts through k-cuts in H.
When there are multiple possible i-cuts, chooses the one that yields the
best division (i.e., the size of the largest sub-problem is the smallest).
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Applying
HINGE+
to Hcg,
k=2
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s13 s14 s 9
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s9 s s
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s11 s17
Width = 5
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
5
Cut Decomposition (CUT)
CUT: A variation of HINGE+.
Input:
A constraint hypergraph H and a maximum cut size k.
Output: An equivalent join tree T of H. The edges of T are labeled.
Process: Finds 1-cuts through k-cuts in H.
CUT guarantees every tree node of T contains at most 2 different cuts.
When there are multiple possible i-cuts that satisfies the condition,
choose the one that yields the best division.
Applying
CUT to
Hcg,
k=2
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Width = 4
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
6
Traverse Decomposition:
TRAVERSE-I
TRAVERSE-I
Input:
A constraint hypergraph H = (V, E) and a set F  E.
Output: An equivalent join tree T of H.
Process: Sweep through the constraint hypergraph from F.
Applying
TRAVERSE-I
to Hcg,
F = {s1}
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Width = 3
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
7
TRAVERSE-II
TRAVERSE-II
Input:
A constraint hypergraph H = (V, E) and two sets F1, F2  E.
Output: An equivalent join tree T of H.
Process: Sweep through the constraint hypergraph from F1 to F2.
Applying
TRAVERSE-II
to Hcg,
F1 = {s1, s2}
F2 = {s9, s16}
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Width = 3
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
8
Cut-and-Traverse Decomposition
(CaT)
CaT: A combination of CUT and TRAVERSE.
Input:
A constraint hypergraph H and a maximum cut size k.
Output: An equivalent join tree T of H.
Process: (1) Apply CUT to H and get a join tree Tm
(2) For every tree node Ni inTm ,
a. If it contains no cut, apply TRAVERSE-I to Ni from an
arbitrary hyperedge.
b. If it contains one cut C1, apply TRAVERSE-I to Ni fromC1.
c. If it contains two cuts C1 and C2, apply TRAVERSE-II from C1 to C2.
Applying CaT
to Hcg,
K = 2.
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Width = 2
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
9
Preliminary Experiments
# Constraints = 20, Maximum arity = 4.
For HINGE+, CUT, and CaT, maximum cut size is 2.
Average CPU time
(millseconds)
Comparing CPU time
30.00
25.00
20.00
15.00
10.00
5.00
0.00
HINGE
HINGE+
CUT
CaT
TRAVERSE
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Variable Number
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
10
Preliminary Experiments
Average Width
Comparing width
13.00
11.00
9.00
7.00
5.00
3.00
1.00
-1.00
HINGE
HINGE+
CUT
CaT
TRAVERSE
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Variable Number
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
11
Conclusions
1.
2.
3.
4.
5.
All these decomposition methods can be performed in polynomial
time.
• HINGE+ O(|V||E|k+1)
• CUT O(|V||E|k+1)
• TRAVERSE O(|V||E|2)
• CaT O(|V||E|k+1)
• k is the maximum cut size
HINGE+ strongly generalizes HINGE.
CaT strongly generalizes CUT.
HYPERTREE strongly generalizes HINGE+, CUT, TRAVERSE,
and CaT.
CaT experimentally decomposes better than HINGE, HINGE+,
CUT, and TRAVERSE.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
12
Future Work
•
•
More thorough experiments on randomly
generated constraint hypergraphs.
Compare CaT with HINGETCLUSTER and
HINGEHYPERTREE.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
13
References
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Gottlob, G., Leone, N., Scarcello, F. : On Tractable Queries and Constraints. In: 10th International Conference
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Decther, R.: Constraint Processing. Morgan Kaufmann (2003)
Freuder, E.C.: A Sufficient Condition for Backtrack-Bounded Search. JACM 32 (4) (1985)
Dechter, R.: Constraint networks, Encyclopedia of Artificial Intelligence, 2nd edition, Wiley. New York, PP.276285. (1992)
Gyssens, M., Jeavons, P.G., Cohen, D.A.: Decomposing Constraint Satisfaction Problems using Database
Techniques. Artificial Intelligence 38 (1989)
Jeavons, P.G., Cohen, D.A., Gyssens, M. : A structural Decomposition for Hypergraphs. Contemporary
Mathematics 178 (1994)
Dechter, R., Pearl. J: Network based heuristic for constraint satisfaction problems, Artificial Intelligence 34 (1)
pp 1-38. (1988)
Decther, R., Pearl. J: Tree Clustering for Constraint Networks. Artificial Intelligence 38 (1998) Gottlob, G.,
Leone, N., Scarcello, F.: Hypertree Decompositions and Tractable Queries. Journal of Computer and System
Sciences 64. (2002)
Robertson, N., Seymour, P.D., Graph Minors II. Algorithmic aspects of tree width, J. Algorithms 7 309-322.
(1986)
Harvey, P., Ghose, A.: Reducing Redundancy in the Hypertree Decomposition Scheme. IEEE International
Conference on Tools with Artificial Intelligence (ICTAI 03). (2003)
Gottlob, G., Leone, N., Scarcello, F.: A comparison of Structural CSP Decomposition Methods. Artificial
Intelligence 124 (2000)
Gottlob, G., Hutle, M., Wotawa, F.: Combining Hypertree, Bicomp, And Hinge Decomposition. ECAI 02 (2002)
Zheng, Y., Choueiry B.Y.: Cut-and-Traverse: A New Structural Decomposition Strategy for Finite Constraint
Satisfaction Problems. CSCLP 04 (2004).
This research is supported by CAREER Award #0133568 from
the National Science Foundation.
Constraint Systems Laboratory
2004-10-13
Yaling Zheng
14